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Properties

Label 4-3024e2-1.1-c1e2-0-34
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 4·7-s + 9·11-s + 9·13-s − 3·17-s + 19-s − 3·23-s + 25-s + 9·29-s − 16·31-s − 12·35-s + 11·37-s − 3·41-s − 15·43-s − 24·47-s + 9·49-s − 3·53-s + 27·55-s + 24·59-s + 27·65-s − 21·73-s − 36·77-s + 9·83-s − 9·85-s + 9·89-s − 36·91-s + 3·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s + 2.71·11-s + 2.49·13-s − 0.727·17-s + 0.229·19-s − 0.625·23-s + 1/5·25-s + 1.67·29-s − 2.87·31-s − 2.02·35-s + 1.80·37-s − 0.468·41-s − 2.28·43-s − 3.50·47-s + 9/7·49-s − 0.412·53-s + 3.64·55-s + 3.12·59-s + 3.34·65-s − 2.45·73-s − 4.10·77-s + 0.987·83-s − 0.976·85-s + 0.953·89-s − 3.77·91-s + 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.787624378\)
\(L(\frac12)\) \(\approx\) \(3.787624378\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_i
11$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.11.aj_bm
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.13.aj_bo
17$C_2^2$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_u
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ba
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.37.al_dg
41$C_2^2$ \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_bs
43$C_2^2$ \( 1 + 15 T + 118 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.43.p_eo
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_abs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.71.a_afa
73$C_2^2$ \( 1 + 21 T + 220 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.73.v_im
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 - 9 T + 116 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.89.aj_em
97$C_2^2$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.97.ad_dw
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.965137907048263139108460575768, −8.589008449866095544237034223878, −8.464932590734321842621488931814, −7.911972630923380182986207603819, −7.07801614677833901460427683945, −6.77068101348321959252142721420, −6.60361958078032421035105611869, −6.26594884693166111284784413966, −5.91917363762003849942154003042, −5.90940285426492562297835219512, −5.13862000974429832960731417428, −4.55742354540754394443417526881, −4.06115362055983740811594091855, −3.62294625780427235763309904170, −3.41358436955173651121436497908, −3.12282023439284920043180058684, −2.04660191587391889708819330559, −1.75039249150844536994762997677, −1.38100492058165702293922969723, −0.63160167226513536419312893279, 0.63160167226513536419312893279, 1.38100492058165702293922969723, 1.75039249150844536994762997677, 2.04660191587391889708819330559, 3.12282023439284920043180058684, 3.41358436955173651121436497908, 3.62294625780427235763309904170, 4.06115362055983740811594091855, 4.55742354540754394443417526881, 5.13862000974429832960731417428, 5.90940285426492562297835219512, 5.91917363762003849942154003042, 6.26594884693166111284784413966, 6.60361958078032421035105611869, 6.77068101348321959252142721420, 7.07801614677833901460427683945, 7.911972630923380182986207603819, 8.464932590734321842621488931814, 8.589008449866095544237034223878, 8.965137907048263139108460575768

Graph of the $Z$-function along the critical line